diff --git a/subroutines/src/pcs/multilinear_kzg/batching.rs b/subroutines/src/pcs/multilinear_kzg/batching.rs
index 17b855b..c235c2b 100644
--- a/subroutines/src/pcs/multilinear_kzg/batching.rs
+++ b/subroutines/src/pcs/multilinear_kzg/batching.rs
@@ -13,9 +13,7 @@ use crate::{
     poly_iop::{prelude::SumCheck, PolyIOP},
     IOPProof,
 };
-use arithmetic::{
-    build_eq_x_r_vec, fix_last_variables, DenseMultilinearExtension, VPAuxInfo, VirtualPolynomial,
-};
+use arithmetic::{build_eq_x_r_vec, DenseMultilinearExtension, VPAuxInfo, VirtualPolynomial};
 use ark_ec::{AffineCurve, PairingEngine, ProjectiveCurve};
 use ark_std::{end_timer, log2, start_timer, One, Zero};
 use std::{marker::PhantomData, rc::Rc};
@@ -38,10 +36,11 @@ where
 /// Steps:
 /// 1. get challenge point t from transcript
 /// 2. build eq(t,i) for i in [0..k]
-/// 3. build \tilde g(i, b) = eq(t, i) * f_i(b)
-/// 4. compute \tilde eq
-/// 5. run sumcheck on \tilde eq * \tilde g(i, b)
-/// 6. build g'(a2) where (a1, a2) is the sumcheck's point
+/// 3. build \tilde g_i(b) = eq(t, i) * f_i(b)
+/// 4. compute \tilde eq_i(b) = eq(b, point_i)
+/// 5. run sumcheck on \sum_i=1..k \tilde eq_i * \tilde g_i
+/// 6. build g'(X) = \sum_i=1..k \tilde eq_i(a2) * \tilde g_i(X) where (a2) is
+/// the sumcheck's point 7. open g'(X) at point (a2)
 pub(crate) fn multi_open_internal<E, PCS>(
     prover_param: &PCS::ProverParam,
     polynomials: &[PCS::Polynomial],
@@ -64,7 +63,6 @@ where
     let num_var = polynomials[0].num_vars;
     let k = polynomials.len();
     let ell = log2(k) as usize;
-    let merged_num_var = num_var + ell;
 
     // challenge point t
     let t = transcript.get_and_append_challenge_vectors("t".as_ref(), ell)?;
@@ -72,40 +70,39 @@ where
     // eq(t, i) for i in [0..k]
     let eq_t_i_list = build_eq_x_r_vec(t.as_ref())?;
 
-    // \tilde g(i, b) = eq(t, i) * f_i(b)
+    // \tilde g_i(b) = eq(t, i) * f_i(b)
     let timer = start_timer!(|| format!("compute tilde g for {} points", points.len()));
-    let mut tilde_g_eval = vec![E::Fr::zero(); 1 << (ell + num_var)];
-    let block_size = 1 << num_var;
+    let mut tilde_gs = vec![];
     for (index, f_i) in polynomials.iter().enumerate() {
+        let mut tilde_g_eval = vec![E::Fr::zero(); 1 << num_var];
         for (j, &f_i_eval) in f_i.iter().enumerate() {
-            tilde_g_eval[index * block_size + j] = f_i_eval * eq_t_i_list[index];
+            tilde_g_eval[j] = f_i_eval * eq_t_i_list[index];
         }
+        tilde_gs.push(Rc::new(DenseMultilinearExtension::from_evaluations_vec(
+            num_var,
+            tilde_g_eval,
+        )));
     }
-    let tilde_g = Rc::new(DenseMultilinearExtension::from_evaluations_vec(
-        merged_num_var,
-        tilde_g_eval,
-    ));
     end_timer!(timer);
 
     let timer = start_timer!(|| format!("compute tilde eq for {} points", points.len()));
-    let mut tilde_eq_eval = vec![E::Fr::zero(); 1 << (ell + num_var)];
-    for (index, point) in points.iter().enumerate() {
+    let mut tilde_eqs = vec![];
+    for point in points.iter() {
         let eq_b_zi = build_eq_x_r_vec(point)?;
-        let start = index * block_size;
-        tilde_eq_eval[start..start + block_size].copy_from_slice(eq_b_zi.as_slice());
+        tilde_eqs.push(Rc::new(DenseMultilinearExtension::from_evaluations_vec(
+            num_var, eq_b_zi,
+        )));
     }
-    let tilde_eq = Rc::new(DenseMultilinearExtension::from_evaluations_vec(
-        merged_num_var,
-        tilde_eq_eval,
-    ));
     end_timer!(timer);
 
     // built the virtual polynomial for SumCheck
-    let timer = start_timer!(|| format!("sum check prove of {} variables", num_var + ell));
+    let timer = start_timer!(|| format!("sum check prove of {} variables", num_var));
 
     let step = start_timer!(|| "add mle");
-    let mut sum_check_vp = VirtualPolynomial::new(num_var + ell);
-    sum_check_vp.add_mle_list([tilde_g.clone(), tilde_eq], E::Fr::one())?;
+    let mut sum_check_vp = VirtualPolynomial::new(num_var);
+    for (tilde_g, tilde_eq) in tilde_gs.iter().zip(tilde_eqs.into_iter()) {
+        sum_check_vp.add_mle_list([tilde_g.clone(), tilde_eq], E::Fr::one())?;
+    }
     end_timer!(step);
 
     let proof = match <PolyIOP<E::Fr> as SumCheck<E::Fr>>::prove(&sum_check_vp, transcript) {
@@ -120,15 +117,23 @@ where
 
     end_timer!(timer);
 
-    // (a1, a2) := sumcheck's point
-    let step = start_timer!(|| "open at a2");
-    let a1 = &proof.point[num_var..];
+    // a2 := sumcheck's point
     let a2 = &proof.point[..num_var];
-    end_timer!(step);
 
-    // build g'(a2)
+    // build g'(X) = \sum_i=1..k \tilde eq_i(a2) * \tilde g_i(X) where (a2) is the
+    // sumcheck's point \tilde eq_i(a2) = eq(a2, point_i)
     let step = start_timer!(|| "evaluate at a2");
-    let g_prime = Rc::new(fix_last_variables(&tilde_g, a1));
+    let mut g_prime_evals = vec![E::Fr::zero(); 1 << num_var];
+    for (tilde_g, point) in tilde_gs.iter().zip(points.iter()) {
+        let eq_i_a2 = eq_eval(a2, point)?;
+        for (j, &tilde_g_eval) in tilde_g.iter().enumerate() {
+            g_prime_evals[j] += tilde_g_eval * eq_i_a2;
+        }
+    }
+    let g_prime = Rc::new(DenseMultilinearExtension::from_evaluations_vec(
+        num_var,
+        g_prime_evals,
+    ));
     end_timer!(step);
 
     let step = start_timer!(|| "pcs open");
@@ -150,8 +155,8 @@ where
 /// Steps:
 /// 1. get challenge point t from transcript
 /// 2. build g' commitment
-/// 3. ensure \sum_i eq(t, <i>) * f_i_evals matches the sum via SumCheck
-/// verification 4. verify commitment
+/// 3. ensure \sum_i eq(a2, point_i) * eq(t, <i>) * f_i_evals matches the sum
+/// via SumCheck verification 4. verify commitment
 pub(crate) fn batch_verify_internal<E, PCS>(
     verifier_param: &PCS::VerifierParam,
     f_i_commitments: &[Commitment<E>],
@@ -175,34 +180,33 @@ where
 
     let k = f_i_commitments.len();
     let ell = log2(k) as usize;
-    let num_var = proof.sum_check_proof.point.len() - ell;
+    let num_var = proof.sum_check_proof.point.len();
 
     // challenge point t
     let t = transcript.get_and_append_challenge_vectors("t".as_ref(), ell)?;
 
-    // sum check point (a1, a2)
-    let a1 = &proof.sum_check_proof.point[num_var..];
+    // sum check point (a2)
     let a2 = &proof.sum_check_proof.point[..num_var];
 
     // build g' commitment
-    let eq_a1_list = build_eq_x_r_vec(a1)?;
     let eq_t_list = build_eq_x_r_vec(t.as_ref())?;
 
     let mut g_prime_commit = E::G1Affine::zero().into_projective();
-    for i in 0..k {
-        let tmp = eq_a1_list[i] * eq_t_list[i];
+
+    for (i, point) in points.iter().enumerate() {
+        let eq_i_a2 = eq_eval(a2, point)?;
+        let tmp = eq_i_a2 * eq_t_list[i];
         g_prime_commit += &f_i_commitments[i].0.mul(tmp);
     }
 
     // ensure \sum_i eq(t, <i>) * f_i_evals matches the sum via SumCheck
-    // verification
     let mut sum = E::Fr::zero();
     for (i, &e) in eq_t_list.iter().enumerate().take(k) {
         sum += e * proof.f_i_eval_at_point_i[i];
     }
     let aux_info = VPAuxInfo {
         max_degree: 2,
-        num_variables: num_var + ell,
+        num_variables: num_var,
         phantom: PhantomData,
     };
     let subclaim = match <PolyIOP<E::Fr> as SumCheck<E::Fr>>::verify(
@@ -219,11 +223,7 @@ where
             ));
         },
     };
-    let mut eq_tilde_eval = E::Fr::zero();
-    for (point, &coef) in points.iter().zip(eq_a1_list.iter()) {
-        eq_tilde_eval += coef * eq_eval(a2, point)?;
-    }
-    let tilde_g_eval = subclaim.expected_evaluation / eq_tilde_eval;
+    let tilde_g_eval = subclaim.expected_evaluation;
 
     // verify commitment
     let res = PCS::verify(